quantum field theory - How to prove useful property of logarithm of generating functional in QFT?

How to prove that $\ln(Z(J))$ generates only connected Feynman diagrams? I can't find the proof of this statement, and have only met its demonstrations for case of 2- and 4-point.

Answer

Assume that the generating functional is given by a sum of all possible diagrams, i.e.

$$Z(J)=\sum_{n_i} D_{n_i}.$$

Furthermore, assume that each diagram D is given by a product of connected diagrams $C_i$, i.e. a diagram D can be disconnected. We will write this as

$$D_{n_i}=\prod_i\frac{1}{n_i!}C_i^{n_i},$$

where dividing by $n_i!$ amounts for a symmetry factor coming from exchanges of propagators and vertices between different diagrams. Combining this with our first expression, we get

$$Z(J)=\sum_{n_i}\prod_i\frac{1}{n_i!}C_i^{n_i}.$$

With some manipulation, this can be shown to be equivalent to

$$Z(J)=\exp\left(\sum_i C_i\right).$$

Taking the logarithm on both sides gives you the desired expression.